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Combined axial and bending loads Interaction equations Design approaches for beam-columns
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Engr. Gabriel Gamana, M.Sc.
2 1.0 Introduction 2.0 Tension Members 3.0 Compression Members 4.0 Beams 5.0 Beam-Columns 6.0 Connections
3 5.1 Introduction 5.2 Interaction Formulas
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6 For many structural members, however, there will be a significant amount of both effects, and such members are called beam–columns.
7 5.2.1 NSCP 2001 Requirement 5.2.1.1 Axial Compression 𝐹𝑜𝑟 𝑓 /𝐹 ≤ 0. 𝑓 𝐹
Strength interaction criterion 𝑓 0.60𝐹௬
8 Stability interaction criterion 𝑓 𝐹
௫^ ᇱ^
௬^ ᇱ^
ଶ
13 5.2.2.1 Braced Versus Unbraced Frames There are two types of secondary moments: P-d (caused by member deflection) and P-Δ (caused by the effect of sway when the member is part of an unbraced frame [moment frame]). 𝑀 = 𝐵ଵ𝑀௧ + 𝐵ଶ𝑀௧ Where; 𝑀 = Required moment strength = 𝑀௨ for LRFD = 𝑀 for ASD 𝑀௧ = Maximum moment assuming that no sidesway occurs, whether the frame is actually braced or not
14 Where; 𝑀௧ = Maximum moment caused by sidesway 𝐵ଵ = amplification factor for the moments occurring in the member when it is braced against sidesway (P-d moments). 𝐵ଶ = amplification factor for the moments resulting from sidesway (P-Δ moments).
15 In addition to the required moment strength, the required axial strength must account for second-order effects. The required axial strength is affected by the displaced geometry of the structure during loading. 𝑃 = 𝑃௧ + 𝐵ଶ𝑃௧ Where; 𝑃௧ = axial load corresponding to the braced condition 𝑃௧ = axial load corresponding to the sidesway condition
16 5.2.2.1.1 Members in Braced Frames The amplification factor given here was derived for a member braced against sidesway—that is, one whose ends cannot translate with respect to each other. 𝐵ଵ =
ଵ
Where; 𝑃 = Required unamplified axial comp. strength = 𝑃௨ for LRFD and 𝑃 for ASD = 𝑃௧ + 𝑃௧ 𝛼 = 1.00 for LRFD and 1.60 for ASD 𝑃ଵ = గ మாூ భ మ
17 The factor Cm applies only to the braced condition.
M 1 /M 2 is a ratio of the bending moments at the ends of the member. M 1 is the end moment that is smaller in absolute value, M 2 is the larger, and the ratio is positive for members bent in reverse curvature and negative for single-curvature bending
18
The factor Ψ has been evaluated for several common situations and is given in Commentary Table C-A-8.1.
19 5.2.2.1.2 Members in Unbraced Frames For a beam–column whose ends are free to translate 𝐵ଶ =
Where; 𝑃௦௧௬ = sum of required load capacities for all columns in the story under consideration 𝑃 ௦௧௬ = total elastic buckling strength of the story under consideration
20 This story buckling strength may be obtained by a sidesway buckling analysis or as 𝑃 ௦௧௬ = 𝑅ெ
Where; 𝑅ெ = 1 − 0.15 (^) ೞೝ 𝑃 = sum of vertical loads in all columns in the story that are part of moment frames 𝐿 = story height ∆ு = interstory drift 𝐻 = story shear (sum of all horizontal forces causing)
